Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking

TL;DR

This paper develops a theoretical framework to classify and understand topological defects and textures in 2D quantum systems with spontaneous symmetry breaking, revealing their connection to topological order and fractional statistics.

Contribution

It introduces a classification scheme for topological defects in 2D quantum orders, linking defect properties to symmetry-enriched topological order and the inflation-restriction sequence.

Findings

Defects relate to deconfined quantum criticality without intrinsic topological order.

Point defects can permute anyons in topologically ordered states.

Textures like skyrmions carry fractional statistics and quantum numbers.

Abstract

We consider two-dimensional (2d) quantum many-body systems with long-range orders, where the only gapless excitations in the spectrum are Goldstone modes of spontaneously broken continuous symmetries. To understand the interplay between classical long-range order of local order parameters and quantum order of long-range entanglement in the ground states, we study the topological point defects and textures of order parameters in such systems. We show that the universal properties of point defects and textures are determined by the remnant symmetry enriched topological order in the symmetry-breaking ground states with a non-fluctuating order parameter, and provide a classification for their properties based on the inflation-restriction exact sequence. We highlight a few phenomena revealed by our theory framework. First, in the absence of intrinsic topological orders, we show a connection between the symmetry properties of point defects and textures to deconfined quantum criticality. Second, when the symmetry-breaking ground state have intrinsic topological orders, we show that the point defects can permute different anyons when braided around. They can also obey projective fusion rules in the sense that multiple vortices can fuse into an Abelian anyon, a phenomena for which we coin "defect fractionalization". Finally, we provide a formula to compute the fractional statistics and fractional quantum numbers carried by textures (skyrmions) in Abelian topological orders.